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Generalized F-iterated function systems on product of metric spaces

Abstract

In this paper, we start from an F-contraction defined on a metric space X into itself, introduced by Wardowski [Fixed Point Theory Appl. 2012 (2012), doi:10.1186/1687-1812-2012-94], and extend it to the case of mappings defined on the space X I into X endowed with the supremum metric, where I is a set of positive integers. Next, we consider the generalized iterated function systems composed of F-contractions on X I improving some fixed point results from the classical Hutchinson–Barnsley theory of iterated function systems. Some illustrative examples are given.

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References

  1. M. F. Barnsley, Fractals Everywhere. 2nd ed., Academic Press Professional, Boston, MA, 1993.

  2. Barnsley M. F., Vince A.: Real projective iterated function systems. J. Geom. Anal. 22, 1137–1172 (2012)

    MATH  MathSciNet  Article  Google Scholar 

  3. V. Berinde, Iterative Approximation of Fixed Points. Lecture Notes in Math., Springer, Berlin, 2007.

  4. D. Dumitru, Generalized iterated function systems containing Meir-Keeler functions. An. Univ. Bucureşti Mat. 58 (2009), 109–121.

  5. Fernau H.: Infinite iterated function systems. Math. Nachr. 170, 79–91 (1994)

    MATH  MathSciNet  Article  Google Scholar 

  6. J. Hutchinson, Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747.

  7. K. Leśniak, Infinite iterated function systems: A multivalued approach. Bull. Pol. Acad. Sci. Math. 52 (2004), 1–8.

  8. E. Llorens-Fuster, A. Petruşel and J.-C. Yao, Iterated function systems and well-posedness. Chaos Solitons Fractals 41 (2009), 1561–1568.

  9. R. Miculescu, Generalized iterated function systems with place dependent probabilities. Acta Appl. Math. 130 (2014), 135–150.

  10. A. Mihail, Recurrent iterated functions systems. Rev. Roumaine Math. Pures Appl. 53 (2008), 43–53.

  11. A. Mihail and R. Miculescu, Applications of fixed point theorems in the theory of generalized IFS. Fixed Point Theory Appl. 2008 (2008), doi:10.1155/2008/312876.

  12. A. Mihail and R. Miculescu, Generalized IFSs on noncompact spaces. Fixed Point Theory Appl. 2010 (2010), doi:10.1155/2010/584215.

  13. N.-A. Secelean, Countable Iterated Function Systems. Lambert Academic Publishing, 2013.

  14. N.-A. Secelean, Generalized countable iterated function systems. Filomat 25 (2011), 21–36.

  15. N.-A. Secelean, Generalized iterated function systems on the space l (X). J. Math. Anal. Appl. 410 (2014), 847–858.

  16. N.-A. Secelean, Iterated function systems consisting of F-contractions. Fixed Point Theory Appl. 2013 (2013), doi:10.1186/1687-1812-2013-277.

  17. N.-A. Secelean, The existence of the attractor of countable iterated function systems. Mediterr. J. Math. 9 (2012), 61–79.

  18. F. Strobin and J. Swaczyna, On a certain generalisation of the iterated function system. Bull. Aust. Math. Soc. 87 (2013), 37–54.

  19. Vince A.: Möbius iterated function systems. Trans. Amer. Math. Soc. 365, 491–509 (2013)

    MATH  MathSciNet  Article  Google Scholar 

  20. D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012 (2012), doi:10.1186/1687-1812- 2012-94.

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Correspondence to Nicolae-Adrian Secelean.

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Secelean, NA. Generalized F-iterated function systems on product of metric spaces. J. Fixed Point Theory Appl. 17, 575–595 (2015). https://doi.org/10.1007/s11784-015-0235-2

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  • DOI: https://doi.org/10.1007/s11784-015-0235-2

Mathematics Subject Classification

  • Primary 28A80
  • Secondary 47H10
  • 54E50

Keywords

  • F-contraction
  • generalized F-contraction
  • generalized iterated function system
  • attractor
  • Hausdorff–Pompeiu metric
  • fixed point